Finite Group Databasebeta

← #27174367

Subgroups — #27174367

Order 12928.

696 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic101C211yesmaximal, characteristic, complemented
2C2×D4×Dic101C212yesmaximal, complemented
3C2×#15692211C211yesmaximal, characteristic, complemented
4#20265109C211yesmaximal, characteristic, complemented
5#20265102C211yesmaximal, characteristic, complemented
6#20265100C212yesmaximal, complemented
7#20265091C212yesmaximal, complemented
8#20265091C211yesmaximal, characteristic, complemented
9#20265028C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic101C212yesmaximal, complemented
11#392×C101C211yesmaximal, characteristic
12C23×Dic101C2212yes—
13C23×Dic101C2211yescharacteristic
14D4×Dic101—28no—
15D4×Dic101—24no—
16#15692211—24no—
17C2×C4⋊Dic101C2212yes—
18C2×C4⋊Dic101C2211yescharacteristic
19C2×#13108053C2212yes—
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×C22⋊Dic101C2212yes—
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C22⋊Dic101C2212yes—
25C2×C22⋊Dic101C2212yes—
26C2×C22⋊Dic101C2212yes—
27C2×C4×Dic101C2211yescharacteristic
28C2×C4×Dic101C2212yes—
29C2×C4×Dic101C2211yescharacteristic
30#15692162C2212yes—
31#15692162C2212yes—
32#15692162C2211yescharacteristic
33C22×D4×C101C2211yescharacteristic
34C22×D4×C101C2212yes—
35C2×C22⋊C4×C101C2212yes—
36C2×C22⋊C4×C101C2211yescharacteristic
37#95×C101C2211yescharacteristic
38C22×Dic101—22no—
39C22×Dic101—22no—
40C22×Dic101—24no—
41C22×Dic101—24no—
42C22×Dic101—24no—
43C22×Dic101D414yescomplemented
44C22×Dic101C2312yes—
45C22×Dic101C2312yes—
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2311yescharacteristic
48C22×Dic101C2311yescharacteristic
49C22×Dic101C2311yescharacteristic
50C4⋊Dic101—22no—
51C4⋊Dic101—24no—
52#13108053—22no—
53#13108053—24no—
54#13108053—22no—
55C22⋊Dic101—24no—
56C22⋊Dic101—24no—
57C22⋊Dic101—22no—
58C22⋊Dic101—24no—
59C22⋊Dic101—24no—
60C4×Dic101—22no—
61C4×Dic101—24no—
62C4×Dic101D414yescomplemented
63C24×C101C2311yescharacteristic
64C24×C101C2312yes—
65C2×D4×C101—24no—
66C2×D4×C101—24no—
67C2×D4×C101—24no—
68C2×D4×C101—22no—
69C2×D4×C101—24no—
70C22×C4×C101C2311yescharacteristic
71C22×C4×C101C2311yescharacteristic
72C22×C4×C101C2312yes—
73C22⋊C4×C101—22no—
74C22⋊C4×C101—24no—
75C23×C101—21no—
76C23×C101—22no—
77C23×C101—22no—
78C23×C101—22no—
79C23×C101—22no—
80C23×C101—22no—
81C23×C101—22no—
82C23×C101—22no—
83C23×C101—21no—
84C23×C101—22no—
85C23×C101C2×D412yes—
86C23×C101#4114yes—
87C23×C101C2411yescharacteristic
88D4×C101—28no—
89D4×C101—216no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—21no—
94C2×C4×C101—22no—
95C2×C4×C101C2×D412yes—
96C2×C4×C101#4114yes—
97C2×Dic101—21no—
98C2×Dic101—22no—
99C2×Dic101—21no—
100C2×Dic101—28no—
101C2×Dic101—22no—
102C2×Dic101—21no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101—22no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101—48no—
110C2×Dic101C2×D412yes—
111C2×Dic101C2×D412yes—
112C2×Dic101C2×D412yes—
113C2×Dic101C2×D412yes—
114C2×Dic101#4112yes—
115C22×C101—22no—
116C22×C101—24no—
117C22×C101—24no—
118C22×C101—24no—
119C22×C101—24no—
120C22×C101—42no—
121C22×C101—41no—
122C22×C101—42no—
123C22×C101—42no—
124C22×C101—42no—
125C22×C101—41no—
126C22×C101—42no—
127C22×C101C22×D411yescharacteristic
128C22×C101C22×D411yescharacteristic
129C22×C101C2×#4111yescharacteristic
130C22×C101C2×#4112yes—
131C22×C101C2×#4112yes—
132C4×C101—22no—
133C4×C101—24no—
134C4×C101—41no—
135Dic101—22no—
136Dic101—22no—
137Dic101—42no—
138Dic101—42no—
139Dic101—41no—
140Dic101#12114yes—
141C2×C101—42no—
142C2×C101—44no—
143C2×C101C2×#12111yescharacteristic
144C2×C101#52511yescharacteristic
145C2×C101#52512yes—
146C2×C101#52912yes—
147C2×C101#53211yescharacteristic
148#2392—1011nomaximal, Sylow
149C101#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1011no—
151C2×C4×D4—1012no—
152C2×#117—1011no—
153#398—1011no—
154#397—1011no—
155#396—1012no—
156#394—1012no—
157#394—1011no—
158#392D10111yescharacteristic
159C4×C4⋊C4—1011no—
160C4×C22⋊C4—1012no—
161C22×D4C2×D10111yescharacteristic
162C22×D4C2×D10112yes—
163C23×C4—1011no—
164C23×C4—1012no—
165C4×D4—2024no—
166C4×D4—2028no—
167#117—2024no—
168C2×C4⋊C4—1011no—
169C2×C4⋊C4—1011no—
170C2×C4⋊C4—1012no—
171C2×C4⋊C4—1011no—
172C2×C4⋊C4—1012no—
173C2×C22⋊C4—1011no—
174C2×C22⋊C4—1012no—
175C2×C22⋊C4—1012no—
176C2×C22⋊C4—1012no—
177C2×C22⋊C4—1012no—
178C2×C22⋊C4C2×D10111yescharacteristic
179C2×C22⋊C4C2×D10112yes—
180C2×C42—1011no—
181C2×C42—1011no—
182C2×C42—1012no—
183#95—1012no—
184#95—1012no—
185#95—1011no—
186#95C2×D10111yescharacteristic
187C24C22×D10111yescharacteristic
188C24C22×D10112yes—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—24no—
194C22×C4—1011no—
195C22×C4—1014no—
196C22×C4—1011no—
197C22×C4—1011no—
198C22×C4—1012no—
199C22×C4—1011no—
200C22×C4—1012no—
201C22×C4—2024no—
202C22×C4—2024no—
203C22×C4—2024no—
204C22×C4—2022no—
205C22×C4—2022no—
206C22×C4C22×D10111yescharacteristic
207C22×C4C22×D10111yescharacteristic
208C22×C4C22×D10112yes—
209C4⋊C4—2022no—
210C4⋊C4—2024no—
211C4⋊C4—2022no—
212C4⋊C4—2022no—
213C4⋊C4—2024no—
214C22⋊C4—22no—
215C22⋊C4—24no—
216C22⋊C4—2022no—
217C22⋊C4—2024no—
218C22⋊C4—2024no—
219C22⋊C4—2024no—
220C22⋊C4—2024no—
221C42—1014no—
222C42—2022no—
223C42—2024no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23D4×D10112yes—
235C23#1310807314yes—
236C23C23×D10111yescharacteristic
237D4—28no—
238D4—216no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—1012no—
245C2×C4—1012no—
246C2×C4—1012no—
247C2×C4—1012no—
248C2×C4—1012no—
249C2×C4—4048no—
250C2×C4—2028no—
251C2×C4—2022no—
252C2×C4—2021no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—2022no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2021no—
260C2×C4—2021no—
261C2×C4—2022no—
262C2×C4D4×D10112yes—
263C2×C4#1310807314yes—
264C22—22no—
265C22—24no—
266C22—24no—
267C22—24no—
268C22—24no—
269C22—42no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—42no—
276C22C2×D4×D10111yescharacteristic
277C22C2×D4×D10111yescharacteristic
278C22#41×D10111yescharacteristic
279C22C2×#1310807312yes—
280C22C2×#1310807312yes—
281C4—22no—
282C4—24no—
283C4—41no—
284C4—1014no—
285C4—4042no—
286C4—4042no—
287C4—4041no—
288C4—2022no—
289C4—2022no—
290C2—42no—
291C2—44no—
292C2#121×D10111yescharacteristic
293C2#2026568311yescharacteristic
294C2#2026568312yes—
295C2#2026570812yes—
296C2#2026572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting, prime radical#392×C101C2
hypercenter#392D101
hypocenterC101#2392
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #392

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×D4×Dic101C2semidirect
2. C2×D4×Dic101C2semidirect
3. C2×#15692211C2semidirect
4. #20265109C2semidirect
5. #20265102C2semidirect
6. #20265100C2semidirect
7. #20265091C2semidirect
8. #20265091C2semidirect
9. #20265028C2semidirect
10. C22⋊C4×Dic101C2semidirect
43. C22×Dic101D4semidirect
62. C4×Dic101D4semidirect
149. C101#2392semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D4×Dic1012
C2×#156922112
#202651092
#202651022
#202651002
#202650912
#202650282
C22⋊C4×Dic1012
#392×C1012
#2392101